Semigroup Forum

, Volume 89, Issue 1, pp 77–104 | Cite as

On semigroups of endomorphisms of a chain with restricted range

  • Vítor H. Fernandes
  • Preeyanuch Honyam
  • Teresa M. Quinteiro
  • Boorapa Singha


Let X be a finite or infinite chain and let \({\mathcal{O}}(X)\) be the monoid of all endomorphisms of X. In this paper, we describe the largest regular subsemigroup of \({\mathcal{O}}(X)\) and Green’s relations on \({\mathcal{O}}(X)\). In fact, more generally, if Y is a nonempty subset of X and \({\mathcal{O}}(X,Y)\) is the subsemigroup of \({\mathcal{O}}(X)\) of all elements with range contained in Y, we characterize the largest regular subsemigroup of \({\mathcal{O}}(X,Y)\) and Green’s relations on \({\mathcal{O}}(X,Y)\). Moreover, for finite chains, we determine when two semigroups of the type \({\mathcal {O}}(X,Y)\) are isomorphic and calculate their ranks.


Transformations Order-preserving Restricted range Rank 



This research was mainly carried out during the visit of the second and fourth authors to Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa and Centro de Álgebra da Universidade de Lisboa between August and October 2011.

The authors would like to thank Cláudia and Francisco Coelho for their help in reviewing the text of this paper.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Vítor H. Fernandes
    • 1
    • 2
  • Preeyanuch Honyam
    • 3
  • Teresa M. Quinteiro
    • 2
    • 4
  • Boorapa Singha
    • 5
  1. 1.Departamento de Matemática, Faculdade de Ciências e TecnologiaUniversidade Nova de LisboaCaparicaPortugal
  2. 2.Centro de Álgebra da Universidade de LisboaLisboaPortugal
  3. 3.Department of Mathematics, Faculty of ScienceChiang Mai UniversityChiang MaiThailand
  4. 4.Instituto Superior de Engenharia de LisboaLisboaPortugal
  5. 5.School of Mathematics and Statistics, Faculty of Science and TechnologyChiang Mai Rajabhat UniversityChiang MaiThailand

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